Abstract:
In the wake of decoupling and linearization semiconductor
device simulation based on van Roosbroecks's equations requires the solution
of convection-diffusion equations. It is well known that due to the occurrence
of local regions of strong convection standard discretizations do not behave properly.
As an alternative among others, mixed methods have been suggested having
their roots in the dual variational formulation of the convection-diffusion
problem.Their efficient implementation has to make use of Lagrangian multipliers.
In a novel approach we already introduce the multiplier prior to discretizing,
through a process called hybridization. In the sequel we use the resulting variational problem to develop a new discretization
scheme. Next, we outline how to implement a standard mixed scheme and investigate
some of its aspects. Finally, the behaviour of the mixed method is illustrated
by a series of numerical experiments.

Abstract:
In the realm of finite element discretizations of elliptic problems
a basic distinction can be made between conforming and nonconforming methods. The latter rely on approximations
not contained in the space the exact solution lives in. Since additional difficulties are thus introduced nonconforming approaches
usually are a last resort: if satisfactory conforming spaces prove elusive they are given a try. For primal variational problems
a mature theory of nonconforming methods has already been developed. This paper seeks to extend these ideas to the mixed formulation
of second order boundary value problems. A ``patch-test'' is designed as a powerful tool for probing the viability of nonconforming
approximations. In particular the criteria are utterly problem-independent. Consequently they can dispense with any regularity
requirements.

Abstract:
The basic drift-diffusion model for semiconductors gives rise to a system of three coupled
partial differential equations: one potential eqation and two continuity equations. G\ärtner was the first to point out that, using different discretization schemes for both
types of equations, the coupling might not be represented correctly. This paper further
elaborates his ideas: in an abstract setting we explain how a dicretization scheme should
take into account the interaction of the equations. To that end we formulate two {\em
compatibility conditions}. Then we develop a technique for quickly assessing the viability
of a particular scheme. It relies on purely local examinations and a few heuristic rules.
Next we investigate several existing schemes to what extent they meet the compatibility
conditions. We show that only the classical Scharfetter-Gummel scheme passes the
tests. All other methods under examination are rejected.

Abstract:
We are concerned with the efficient solution of saddle point problems arising from
the mixed discretization of 2nd order elliptic problems in two dimensions.
We consider the mixed discretization of the boundary value problem by means of
lowest order Raviart-Thomas elements. This leads to a saddle point problem,
which can be tackled by Uzawa-like iterative solvers. We suggest a prior modification
of the saddle point problem according to the augmented Lagrangian
approach in order to make it more amenable to the iterative
procedure. In order to boost the speed of iterative methods,
we additionally employ a multilevel preconditioner first presented
by Vassilevski and Wang in. It is based on a special splitting of the
space of vector valued fluxes, which exploits the close relationship between
piecewise linear continuous finite element functions and divergence free fluxes.
We prove that this
splitting gives rise to an optimal preconditioner: it achieves condition numbers
bounded independently on the depth of refinement. The proof is set in the framework
of Schwarz methods. It relies on established results about
standard multilevel methods as well as a strengthened Cauchy-Schwarz inequality
for lowest order Raviart-Thomas spaces.

Abstract: The cost-effective design of electronic
microstructures requires an advanced modeling and coupled simulation of various
physical effects. The classical isothermal approach leads to the basic
drift-diffusion model for semiconductor device simulation. In the stationary case, it
represents a coupled nonlinear system consisting of a Poisson equation for the
electricpotential and two continuity equations for the electron and hole flow. We
discuss various discretization schemes with special emphasis on mixed finite element
methods and we further address efficient numerical solution techniques including
adaptive multilevel methods. Finally, to allow for ambient conditions such as
external magnetic fields we consider consistent extensions of the classical model and
discuss perspectives for their numerical treatment.

Abstract: Adaptive multilevel algorithms along with finite element
discretizations are the most po werful tools for the fast and accurate solution of
elliptic boundary value problems over irregular do mains. They involve the delicately
tuned interaction of a posteriori error estimation, local mesh refinement, multilevel
computations and iterative solution.

This introduces unprecedented complexity into mathematical software, in terms of both
sheer size of the code and sophisticated dynamic data structures. Advanced techniques
of software engineering have to be used in order to develop reliable codes quickly.
In particular, we relied on the object oriented paradigm of software design. It
requires us to split the task into logical units and to fix several levels of
abstraction within those units. This leads to a set of so-called classes, arranged in
a tree, which jointly manage data and supply functions.

Crudely speaking, two types
of classes could be identified: Local classes and global classes. The former are
concerned with components of the mesh and the latter deal with the entire mesh. Thus
highly versatile building blocks for adaptive multilevel finite element schemes have
been created. They relieve the user of caring about the internal management of the
grid and numerical data and offer well defined interfaces. They can be quickly
assembled into a code for a vast range of boundary value problems, including systems
of equations and saddle point problems.

Abstract: We are concerned with the efficient computation of
the magnetic field induced by a stationary strictly conservative current. According
to Maxwell's equations the magnetic field has vanishing divergence and its $\curl$
equals the current. The current is assumed to be discretized by means of lowest order
Raviart--Thomas elements. The magnetic field is sought in N'ed'elec's
H(curl)-conforming finite element spaces of order 1. We suggest a multilevel
approach for the solution of this problem. It involves two steps: First a discrete
stream function of the current is determined. This is a direct procedure based on a
multilevel splitting of the current. In a second step a curl-free correction is
computed by means of a multilevel preconditioned CG method to ensure the divergence
free condition. The overall amount of work is proportional to the number of unknown
values of the discrete magnetic field.

Abstract: This work aims to provide a rigorous theoretical
examination of multilevel preconditioning schemes for some discrete variational
problems in three dimensions, which involve the differential operators div and
curl. Such kinds of problems occur, for instance, in the dual formulation of
elliptic boundary value problems, in which case they are posed over subspaces of
H(div) and h(curl).
The investigations are set in a finite element framework relying on simplicial meshes
and the finite element schemes introduced by Raviart and Thomas. We take a fresh
look at the construction of these spaces, viewing them from the angle of differential
forms. Thus, we arrive at a fairly canonical procedure to obtain these particular
finite elements and forge a unified analysis of their properties. In addition, we
managed to establish approximation estimates in fractional Sobolev spaces and new
discrete extension theorems.
Our main focus is on the augmented Lagrangian technique , applied to the saddle point
system arising from the mixed finite element discretization of an ordinary scalar
second order elliptic problem. Uzawa algorithms, the minimal residual method and the
method of Bank, Welfert and Yserentant provide algorithms for its iterative solution.
We show that it takes only an efficient preconditioner for discrete operators related
to the bilinear form (u,v)+r(divu,divv), to achieve
methods of optimal computational complexity. Of course, the preconditioner must not
be adversely affected by large values of the augmented Lagrangian parameter r.
We extend the approach of Vassilevski and Wang in two dimensions to obtain a
multilevel splitting of Raviart--Thomas spaces in 3D, built upon a sequence of nested
triangulations. For the treatment of the crucial divergence free vector fields we
resort to a nodal BPX--type decomposition of N'ed'elec spaces. We discover that with
slight modifications the hierarchical bases scheme of Cai, Goldstein and Pasciak is
covered, as well.
Based on algebraic multilevel theory , we investigate the stability of the
decompositions of lowest order N'ed'elec spaces with respect to the bilinear form
(curl u,curl v). To cope with its nontrivial kernel we switch to the
related quotient space. Under certain assumptions on the regularity of a
curl curl-boundary value problem, duality arguments according to Zhang yield
the stability of the nodal splitting, independent on the number of refinement levels.
By the extension theorem this result carries over to general domain if no boundary
values are imposed.
The findings also show that the direct elimination of the non--solenoidal part of the
flux suggested by Ewing and Wang for 2D applications is just as efficient for mixed
problems in 3D. Also they enable us to construct fast solvers for first order system
least squares discretizations of second order elliptic problems. Moreover, the
results prove useful for designing multilevel schemes for the computation of vector
potentials in magnetostatics.

Abstract: In this paper we consider second order scalar
elliptic boundary value problems posed over three--dimensional domains and their
discretization by means of mixed Raviart--Thomas finite elements. This leads to saddlepoint problems featuring a discrete flux vector field as additional unknown.
Following Ewing and Wang, the proposed solution procedure is based splitting the
flux into divergence free components and a remainder. It leads to variational
problem involving solenoidal Raviart--Thomas vector fields.
A fast iterative solution method for this problem is presented. It exploits the
representation of divergence free vector fields as curls of the
H(curl)-conforming finite element functions introduced by N'ed'elec. We show
that a nodal multilevel splitting of these finite element spaces gives rise to an
optimal preconditioner for the solenoidal variational problem: Duality
techniques in quotient spaces and modern
algebraic multigrid theory are the main tools for the proof.

Abstract:
The mixed variational formulation of many elliptic boundary value problems involves
vector valued function spaces, like, in three dimensions, H(curl) and
H(div). Thus finite element subspaces of these function spaces are indispensable
for effective finite element discretization schemes. Given a simplicial
triangulation of the computational domain, among others, Raviart, Thomas and
N'ed'elec have found suitable conforming finite elements for H(div) and
H(curl), respectively. At first glance, it is hard to detect a common
guiding principle behind these approaches. We take a fresh look at the construction
of the finite spaces viewing them from the angle of differential forms. This is
motivated by the well--known relationships between differential forms and
differential operators: both div, (curl and grad can be regarded as
special incarnations of the exterior derivative of a differential form. Moreover,
in the realm of differential forms most concepts are basically
dimension--independent. Thus, we arrive at a fairly canonical procedure to
construct conforming finite element subspaces of function spaces related to
differential forms. In any dimension we can give a simple characterization of the
local polynomial spaces and degrees of freedom underlying the definition of the
finite element spaces. With unprecedented ease we can recover the familiar H(div)-
and H(curl)-conforming finite elements and establish the unisolvence of
degrees of freedom. In addition, the use of differential forms makes it possible to
establish crucial algebraic properties of the canonical interpolation operators and
representation theorems in a single sweep for all kinds of spaces.

Abstract: A widely used approach for the computation of
time--harmonic electromagnetic fields is based on the well--known double--curl
equation for either E or H. An appealing choice for finite element
discretization are edge elements, the lowest order variant of a
H(curl)-conforming family of finite elements. We end up with a large sparse
linear system of equations, which can only be solved iteratively. In this paper we
focus on fast multilevel preconditioned iterative schemes. Yet, the nullspace of the
curl-operator comprises a considerable part of all spectral modes on the
finite element grid. Thus standard multilevel solvers are rendered inefficient, as
they essentially hinge on smoothing procedures like Gauss--Seidel relaxation, which
cannot provide a satisfactory error reduction for modes with very small or even
negative eigenvalues. A remedy is offered by an extended multilevel algorithm that
relies on corrections in the space of discrete scalar potentials. After every
standard V--cycle with respect to the canonical basis of edge elements, error
components in the nullspace are removed by an additional projection step.
Furthermore, a simple criterion for the coarsest mesh is derived to guarantee both
stability and efficiency of the iterative multilevel solver. For the whole scheme we
observe convergence rates independent of the refinement level of the mesh. The
sequence of nested meshes required for multilevel techniques is constructed by
adaptive refinement. For controlling adaptive mesh refinement we have devised an
a--posteriori error indicator based on stress recovery. }

Abstract:
We are concerned with the design and analysis of a multigrid algorithm for
H(div)-elliptic linear variational problems. The discretization is based on
H(div)-conforming Raviart--Thomas elements. A thorough examination of the
relevant bilinear form reveals that a separate treatment of vector fields
in the kernel of the divergence operator and its complement is paramount.
We exploit the representation of discrete solenoidal vector fields as
$\curl$s of finite element functions in so-called N'ed'elec spaces.
It turns out that a combined nodal multilevel decomposition of both the
Raviart--Thomas and N'ed'elec finite element spaces provides the
foundation for a viable multigrid method. Its Gauß-Seidel smoother involves
an extra stage where solenoidal error components are tackled. By means of
elaborate duality techniques we can show the asymptotic optimality in the
case of uniform refinement. Numerical experiments confirm that the typical
multigrid efficiency is actually achieved for model problems.

Abstract: In this paper we are concerned with the efficient
solution of discrete variational problems related to the bilinear form
(curl.,curl.)+(.,.) defined on H(curl). This is a core task in
the time--domain simulation of electromagnetic fields, if implicit timestepping is
employed. We rely on N'ed'elec's H(curl)-conforming finite elements (edge
elements) to discretize the problem. We construct a multigrid method for the fast
iterative solution of the resulting linear system of equations. Since proper
ellipticity of the bilinear form is confined to the complement of the kernel of the
curl--operator, Helmholtz--decompositions are the key to the design of the
algorithm: Kern(curl) and its complement require a separate treatment. Both can
be tackled by nodal multilevel decompositions where for the former the splitting is
set in the space of discrete scalar potentials. Under certain assumptions on the
computational domain and the material functions a rigorous proof of the asymptotic
optimality of the multigrid method can be given; it shows that convergence does not
deteriorate on very fine grids. The results of numerical exeriments confirm the
practical efficiency of the method.

Abstract:
In the realm of finite element discretizations of elliptic problems a basic
distinction can be made between conforming and nonconforming methods. The latter
rely on approximations not contained in the spaces in which the continuous
variational problem is posed. This paper investigates nonconforming finite element
approximations of the spaces H(div) and H(curl) of vector valued functions.
First, we extend the "generalized patch test" which has been developed to provide
necessary and sufficient conditions for the viability of nonconforming schemes for
standard Sobolev spaces. Then we use the calculus of differential forms to derive
coupling conditions sufficient for success in the patch test. Based on this result
we design convergent nonconforming methods for a number of variational problems
involving the above--mentioned function spaces. Finally, we point out that a naive
approach leads to an inconsistent scheme.

Abstract: This paper is intended as a survey of current results
on algorithmic and theoretical aspects of overlapping Schwarz methods for discrete
H(curl and H(div)-elliptic problems set in suitable finite element spaces. The
emphasis is on a unified framework for the motivation and theoretical study of the
various approaches developed in recent years. Generalized Helmholtz decompositions -
orthogonal decompositions into the null space of the relevant differential operator
and its complement - are crucial in our considerations. It turns out that the
decompositions the Schwarz methods are based upon have to be designed separately for
both components. In the case of the null space, the construction has to rely on
liftings into spaces of discrete potentials. Taking the cue from well-known Schwarz
schemes for second order elliptic problems, we devise uniformly stable splittings of
both parts of the Helmholtz decomposition. They immediately give rise to powerful
preconditioners and iterative solvers.

Abstract: The focus of this paper is on the efficient solution
of boundary value problems involving the double curl-operator. Those arise in
the computation of electromagnetic fields in various settings, for instance when
solving the electric or magnetic wave equation with implicit timestepping, when
tackling time-harmonic problems or in the context of eddy-current computations.
Their discretization is based on N'ed'elec's curl-conforming edge elements on
unstructured grids. In order to capture local effects and to guarantee a prescribed
accuracy of the approximate solution adaptive refinement of the grid controlled by a
posteriori error estimators is employed. The hierarchy of meshes created through
adaptive refinement forms the foundation for the fast iterative solution of the
resulting linear systems by a multigrid method. The guiding principle underlying the
design of both the error estimators and the multigrid method is the separate
treatment of the kernel of the curl-operator and its orthogonal complement.
Only on the latter we have proper ellipticity of the problem. Yet, exploiting the
existence of computationally available discrete potentials for edge element spaces,
we can switch to an elliptic problem in potential space to deal with nullspace of
$\curl$. Thus both cases become amenable to strategies of error estimation and
multigrid solution developed for second order elliptic problems. The efficacy of the
approach is confirmed by numerical experiments which cover several model problems and
an application to waveguide simulation.

Abstract:
We consider H(curl)-elliptic problems that have been discretized by means of
N'ed'elec's edge elements on tetrahedral meshes. Such problems occur
in the numerical compuation of eddy currents. From the defect equation
we derive localized expressions that can be used as a posteriori error
estimators to control adaptive refinement. Under certain assumptions
on material parameters and computational domains, we derive local
lower bounds and a global upper bound for the total error measured in the
energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type
decomposition of the error into an irrotational part and a weakly solenoidal part.

Abstract:
Conforming finite elements in H(div) and H(curl), can be regarded as
discrete differential forms (Whitney--forms). The construction of such forms is
based on an interpolation idea, which boils down to a simple extension of the
differential form to the interior of elements. This flexible approach can
accommodate elements of more complicated shapes than merely tetrahedra and bricks.
The pyramid serves as an example for the successful application of the
construction: New Whitney forms are derived for it.

Abstract:
In the context of macroscopic simulation of electrorheological fluids, we present an
extension of the classical Bingham model. It accommodates arrangements beyond pure
shear flows and will make possible fully three-dimensional simulations.
For the numerical solution of the resulting nonsmooth minimization problem we propose
the method of augmented Lagrangians which turns out to be an appropriate iterative
solver for such problems. Finally, we present computational results
illustrating the electrorheological effect for various shear rates and electric field
strengths in the case of an electrorheological suspension rotating between two
revolving cylinders.

Abstract:
The vector potential of a solenoidal vector field, if it exists, is not unique in
general. Any procedure that aims to determine such a vector potential typically
involves a decision on how to fix it. This is referred to by the term gauging.
Gauging is an important issue in computational electromagnetism, whenever discrete
vector potentials have to be computed. In this paper a new gauging algorithm for
discrete vector potentials is introduced that relies on a hierarchical multilevel
decomposition. With minimum computational effort it yields vector potentials whose
L2-norm does not severely blow up. Thus the new approach compares
favorably to the widely used co-tree gauging.

Abstract:
We consider anisotropic second order elliptic boundary value problems in two
dimensions, for which the anisotropy is exactly aligned with the coordinate axes.
This includes cases where the operator features a singular perturbation in one
coordinate direction, whereas its restriction to the other direction remains neatly
elliptic. Most prominently, such a situation arises when polar coordinates
are introduced.
The common multigrid approach to such problems relies on line relaxation in the
direction of the singular perturbation combined with semi-coarsening in the other
direction. Taking the idea from classical Fourier analysis of multigrid, we employ
eigenspace techniques to separate the coordinate directions. Thus, convergence of
the multigrid method can be examined by looking at one-dimensional operators only.
In a tensor product Galerkin setting, this makes it possible to confirm that the
convergence rates of the multigrid V-cycle are bounded independently of the number
of grid levels involved. In addition, the estimates reveal that convergence is also
robust with respect to a singular perturbation in one coordinate direction.
Finally, we supply numerical evidence that the algorithm performs satisfactorily
in settings more general than those covered by the proof.

Abstract:
Many linear boundary value problems arising in computational physics can be
formulated in the calculus of differential forms. Discrete differential forms
provide a natural and canonical approach to their discretization. However, much
freedom remains concerning the choice of discrete Hodge operators, that is,
discrete analogues of constitutive laws. A generic discrete Hodge operator is
introduced and it turns out that most finite element and finite volume schemes
emerge as its specializations. We reap the possibility of a unified convergence
analysis in the framework of discrete exterior calculus.

Abstract:
We consider the quasi-magnetostatic eddy current problem discretized by means of
lowest order curl-conforming finite elements (edge elements) on
tetrahedral meshes. Bounds for the discretization error in the finite element
solution are desirable to control adaptive mesh refinement. We propose a local
a-posteriori error estimator based on higher order edge elements: The residual
equation is approximately solved in the space of p-hierarchical surpluses. Provided
that a saturation assumption holds, we show that the estimator is both reliable and
efficient.

Abstract:
The focus of this paper is on boundary value problems for Maxwell's equations that
feature cylindrical symmetry both of the domain and the data. Thus, by resorting to
cylindrical coordinates, a reduction to two dimensions is possible.
However, cylindrical coordinates introduce a potentially malicious singularity at
the axis rendering the variational problems degenerate. Ultimately, this cripples
the performance of a standard multigrid solver.
Line relaxation in radial direction and semicoarsening can successfully reign in
the degeneracy. In addition, the lack of strong ellipticity of the double-curl
operator entails using special hybrid smoothing procedures. All these techniques
combined yield a fast multigrid solver.
The theoretical investigation of the method relies on blending generalized Fourier
techniques and modern variational multigrid theory. We first determine invariant
subspaces of the multigrid iteration operator and analyze the smoothers therein.
Under certain assumptions on the material parameters we manage to show uniform
convergence of a symmetric V-cycle.

Abstract:
In this paper a novel symmetric FEM-BEM-coupling for the E-based eddy current
model is derived in a rigorous fashion. To that end the properties of potentials
and boundary integral operators arising from a Stratton-Chu-type representation
formula for the electric field in the non-conducting region are thoroughly analyzed
in a Hilbert-space setting. It yields a variational problem with symmetric bilinear
form that is coercive in the natural function spaces. Unknowns are the electric
field inside the conductor and the equivalent surface current related to the
magnetic field. Existence and uniqueness of solutions, and the convergence of
conforming a finite element -- boundary element Galerkin discretizations
immediately follow. In particular, schemes based on $\curl$-conforming edge
elements and divergence-conforming surface elements are examined.

Abstract:
This paper deals with the simulation of induction heating of conducting workpieces
with a complex shape and topology. We assume that its conductivity is high, which,
owing to the skin effect bars the fields from penetrating deep into the conductors.
This justifies the use of a simple magnetostatic model that describes the magnetic
field outside the workpiece and the inductor. Field concentrating plates can also
be taken into account. After possible holes in the conductors have been patched
with cutting surfaces, a scalar magnetic potential can be employed. The actual
computation relies on a boundary integral equation of the second kind, discretized
by means of piecewise constant boundary elements. Thus we get an approximation of
the surface currents. Then we use the skin effect formula to determine the rate of
heat generation inside the workpiece.

Abstract:
Discrete differential forms provide a natural and canonical approach to the
discretization of many physical quantities. If the solution is sufficiently smooth,
sparse grid finite elements techniques lead to an improved ratio of storage
requirements versus the accuracy of discrete approximations. Interpolation
estimates are proved in the context of Whitney forms. The results for Lagrangian
finite elements emerge as particular case. We address the influence of quadrature
rules used for the evaluation of degrees of freedom.

Abstract:
In [R. Hiptmair, Multigrid method for Maxwell's equations, SIAM
J. Numer. Anal., 36 (1999), pp. 204-225] a novel multigrid method for
discrete H(curl)-elliptic boundary value problems has been proposed.
These frequently occur in computational electromagnetism, in particular in the
context of eddy current simulation.
This paper focuses on the analysis of those nodal multilevel decompositions of the
spaces of edge finite elements that form the foundation of the multigrid method.
This paper provides a significant extension of the existing theory. In particular,
asymptotically uniform convergence of the multigrid method with respect to the
number of refinement levels can be established in a realistic setting for eddy
current problems that features truly non-conducting regions, reentrant corners and
complex topologies of the conductor.
This is made possible by using approximate Helmholtz-decompositions of the function
space H(curl) into an H1-regular subspace and gradients.
The main results of standard multilevel theory for H1-elliptic can then
be applied to both subspaces. This yields preliminary decompositions still beyond
the finite element setting. Judicious alterations can cure this.

Abstract:
The calculus of differential forms can be used to devise a
unified description of discrete differential forms of any order and
polynomial degree on simplicial meshes in any spatial dimension. A
general formula for suitable degrees of freedom is also
available. Fundamental properties of nodal interpolation can be
established easily. It turns out that higher order spaces, including
variants with locally varying polynomial order, emerge from the usual
Whitney-forms by local augmentation. This paves the way for an
adaptive p-version approach to discrete differential forms.

Abstract:
Discrete differential forms should be used to deal with
the discretization of boundary value problems that can be stated in
the calculus of differential forms. This approach preserves the
topological features of the equations. Yet, the discrete
counterparts of the metric-dependent constitutive laws remain
elusive.
I introduce a few purely algebraic constraints that matrices
associated with discrete material laws have to satisfy. It turns out
that most finite element and finite volume schemes comply with these
requirements. Thus convergence analysis can be conducted in a
unified setting. This discloses basic sufficient conditions that
discrete material laws have to meet in order to ensure convergence
in the relevant energy norms.

Abstract:
For a Lipschitz-polyhedron $\Omega\subset\mathbb{R}^3$ we consider eigenvalue
problems curl\alphacurlu
=\lambda\Vu$ and grad\alpha divu
= \lambdau$, $\lambda>0$, set in H(curl) and H(div). They are
= discretized by means of the
conforming finite elements introduced by Nedelec. The preconditioned inverse
iteration in its subspace variant is adapted to these problems. A standard
multigrid scheme serves as preconditioner. The main challenge arises from the large
kernels of the operators curl and div. However, thanks to the choice of
finite element spaces these kernels have a direct representation through the
gradients/rotations of discrete potentials. This makes it possible to use a
multigrid iteration in potential space to obtain approximate projections onto the
orthogonal complements of the kernels. There is ample evidence that this will lead
to an asymptotically optimal method. Numerical experiments confirm the excellent
performance of the method even on very fine grids.

Abstract:
We consider a bounded Lipschitz-polyhedron O of general
topology equipped with a tetrahedral triangulation that induces a mesh
of the surface S. We seek a maximal set of surface edge cycles that
are independent in H1(S) and bounding with respect to the
exterior of O.

We present an algorithm for constructing suitable 1-cycles: First,
representatives of a basis of the homology group H1(S) are
constructed, merely using the combinatorial description of the surface mesh.
Then, a duality pairing based on linking numbers is used to determine
those combinations that are bounding w.r.t. the exterior$. This is
the key to circumventing a triangulation of the exterior region
in the computations. For shape-regular, quasiuniform
families of meshes, the asymptotic complexity of the algorithm is shown to be
O(N^2), where N is the number of edges of the surface mesh.
The scheme provides an essential preprocessing step for novel boundary element
methods in computational electromagnetism, which rely on discrete divergence-free
vectorfields and their description through stream functions.

Abstract:
We consider the electric field integral equation on the surface of polyhedral
domains and its Galerkin-discretization by means of divergence-conforming boundary
elements. With respect to a Hodge-decomposition the continuous variational problem
is shown to be coercive. However, this does not immediately carry over to the
discrete setting, as discrete Hodge decompositions fail to possess essential
regularity properties. Introducing an intermediate semidiscrete Hodge decomposition
we can bridge the gap and come up with asymptotic a-priori error estimates.
Hitherto, those had been elusive for non-smooth boundaries.

Abstract:
We consider the Maxwell equations in a domain with Lipschitz boundary and the
boundary integral oprator A occurring in the Calderon projector. We prove
an inf-sup condition for A using a Hodge decomposition for tangent fields.

We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution.

We then consider Galerkin discretizations based on Raviart-Thomas spaces. We show that these spaces have a discrete Hodge decomposition which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods.

Abstract:
We consider the scattering of electromagnetic waves at a dielectric object with a
rough surface. We investigate the coupling of a weak formulation of Maxwell's
equations inside the scatterer with boundary integral equations that arise from the
homogeneous problem in the unbounded region outside the scatterer. The symmetric
coupling approach based on the full Calderon projector for Maxwell's equations is
employed. By splitting both the electric field inside the scatterer and the surface
currents into components of predominantly electric and magnetic nature, we can
establish coercivity of the coupled variational problem.
Discretization relies on both curl-conforming edge elements inside the
scatterer and div-conforming boundary elements for the surface currents.
The splitting idea adjusted to the discrete setting permits us to show
uniform stability of the discretized problem. We exploit it to come up
with a-priori convergence estimates.

Abstract:
This paper studies numerical methods for eddy current problems in the case of
homogeneous, isotropic, and linear materials. It provides a survey of approaches
that entirely rely on boundary integral equations and their conforming Galerkin
discretization. The pivotal role of potentials is discussed, as well as the
topological issues raised by their use. Direct boundary integral equations and the
so-called symmetric coupling of the integral equations corresponding to the
conductor and the non-conducting regions is employed. It gives rise to coupled
variational problems that are elliptic in suitable trace spaces. This implies
quasi-optimal convergence of Galerkin boundary element schemes.

Abstract:
This article discusses finite element Galerkin schemes for a number of linear
model problems in electromagnetism. The finite element schemes are introduced
as discrete differential forms, matching the coordinate independent statement
of Maxwell's equations in the calculus of differential forms. The asymptotic
convergence of discrete solutions is investigated theoretically. As discrete
differential forms represent a genuine generalisation of conventional
Lagrangian finite elements, the analysis is based upon a judicious adaptation of
established techniques in the theory of finite elements. Risks and
difficulties haunting finite element schemes that do not fit the framework of
discrete differential forms are highlighted.

Abstract:
Local mode analysis is an established technique to get quantitative estimates for
the convergence rates of multigrid for boundary value problems discretized on
regular grids. However, when the discrete differential operators are more
complicated than a five-point stencil in two dimensions, the computations turn out
to be extremely tedious. They are only feasible with the aid of computer software.
The foundation for such a code is laid by the theory and implementation presented
in this paper: We describe a code that can compute multigrid convergence rates for
translation invariant local operators on infinite tensor product grids, the usual
setting of local mode analysis. The user only has to specify operator stencils.

Abstract:
In eddy current problems there is a natural distinction between conducting and
non-conducting regions, in which the equations of the mathematical model have
distinct features. Thus, the issue of coupling discrete models in both regions
arises. It its particularly urgent, when different frames of reference are
used, as in the case of moving conductors and a Lagrangian perspective.
We focus on finite element schemes (FEM) and study primal and dual variational
formulations. It turns out that the coupling of primal and dual formulations is
straightforward even in the case of non-matching meshes across the boundaries
of conductors, because the coupling is completely taken into account in
weak form. This makes primal-dual coupling an attractive option for dealing
with moving conductors in the context of FEM.

Abstract:
The physical nature of electromagnetic fields suggests differential
forms as the natural tool for their mathematical modeling. Co-chains
on cellular complexes offer a discretization that preserves fundamental
topological features of the laws of electromagnetism. However, the
constitutive equations cannot be dealt with in this framework, which
entails extending co-chains to discrete differential forms, whose
lowest order variant is known as Whitney-forms.

Abstract:
Discrete differential forms are a generalization of the common H1-conforming
Lagrangian elements. For the latter Galerkin schemes based on sparse grids are well
known, and so are fast iterative multilevel solvers for the discrete Galerkin
equations. We extend both the sparse grid idea and the design of multilevel methods
to arbitrary discrete differential forms. The focus of this presentation will be on
issues of efficient implementation and numerical studies of convergence of
multigrid solvers.

Abstract:
Many boundary integral equation methods used in the simulation of direct
electromagnetic scattering of a time-harmonic wave at a perfectly conducting
obstacle break down, when applied at frequencies close to a resonant frequency of
the obstacle. A remedy is offered by special indirect boundary element methods
based on the so-called combined field integral equation. However, hitherto no
theoretical results about the convergence of discretized combined field integral
equations have been available.
In this paper we propose a new combined field integral equation, convert it
into variational form, establish its coercivity in the natural trace spaces
for electromagnetic fields, and conclude existence and uniqueness of solutions
for any frequency. Moreover, a conforming Galerkin discretization of the
variational equations by means of $\bDiv$-conforming boundary elements can be shown
to be asymptotically quasi-optimal. This permits us to derive quantitative
convergence rates on sufficiently fine, uniformly shape-regular sequences of
surface triangulations.

Abstract: Finite element approximations of eddy current
problems that are entirely based on the magnetic field H are haunted by the
need to enforce the algebraic constraint curl H=0 in non-conducting
regions. As an alternative to techniques employing combinatorial Seifert (cutting)
surfaces in order to introduce a scalar magnetic potential, we propose mixed
multi-field formulations, which enforce the constraint in the variational
formulation. In light of the fact that the computation of cutting surfaces is
expensive, the mixed finite element approximation is a viable option despite the
increased number of unknowns.

Abstract:
This article presents the mathematical foundation of a symmetric boundary
element method for the computation of eddy currents in a linear homogeneous
conductor which is exposed to an alternating magnetic field. Starting from the
A-based variational formulation of the eddy current equations and a
related transmission problem, the problem inside and outside the conductors can be
reformulated as integral equations on the boundary of the conductors. Surface
currents occur as new unknowns of this direct formulation. The integral equations
can be coupled in a symmetric fashion using the transmission conditions for
A and H. The resulting variational problem is elliptic in
suitable trace spaces.
A conforming Galerkin boundary element discretization is employed, which relies on
surface edge elements and provides quasi-optimal discrete approximations for the
tangential traces of $A and H. Surface stream functions
supplemented with co-homology vector fields ensure the vital zero divergence of the
discrete equivalent surface currents. Simple expressions allow the computation of
approximate total Ohmic losses and surface forces from the discrete boundary data.

Abstract:
A new discrete non-reflecting boundary condition for the time-dependent Maxwell
equations describing the propagation of an electromagnetic wave in an infinite
homogenous lossless rectangular waveguide with perfectly conducting walls is
presented. It is derived from a virtual spatial finite difference discretization
of the problem on the unbounded domain. Fourier transforms are used to decouple
transversal modes. A judicious combination of edge based nodal values
permits us to recover a simple structure in Laplace domain. Using this, it is possible
to approximate the convolution in time by a similar fast convolution algorithm
as for the standard wave equation.

Keywords :
Eddy current model, coupling of circuits and field equations, voltage and current
excitation, variational formulations

Abstract:
This paper describes a novel quadrilateral edge element discretisation of
Maxwell's equations in which the effects of dispersion are minimised.
A
modified edge finite element stencil is proposed and it is subsequently shown
how this can be expressed in terms of new
material coefficients thus allowing us to incorporate both Dirchlet
and Neumann boundary conditions in a natural fashion. To demonstrate the success of the proposed
procedure, we include a series of numerical examples. First we apply the approach
to plane and circular wave propagation problems. Secondly, we apply the
approach to a series of electromagnetic scattering problems. For the
electromagnetic scattering computations, we monitor the
effect of the modified edge finite element stencil on the scattering width
output. We use a $hp$--edge element code as a benchmark for all our electromagnetic
scattering computations.

Abstract:
This paper describes a novel quadrilateral edge element discretisation of
Maxwell's equations in which the effects of dispersion are minimised.
A
modified edge finite element stencil is proposed and it is subsequently shown
how this can be expressed in terms of new
material coefficients thus allowing us to incorporate both Dirchlet
and Neumann boundary conditions in a natural fashion. To demonstrate the success of the proposed
procedure, we include a series of numerical examples. First we apply the approach
to plane and circular wave propagation problems. Secondly, we apply the
approach to a series of electromagnetic scattering problems. For the
electromagnetic scattering computations, we monitor the
effect of the modified edge finite element stencil on the scattering width
output. We use a $hp$--edge element code as a benchmark for all our electromagnetic
scattering computations.

Abstract:
The computation of the resonant frequencies for open and closed cavities is not a
trivial task: Multi-materials and sharp corners all give rise to highly singular
eigenfunctions. However, an approach using hp-finite elements is well suited to such
problems and, with the correct combination of h- and p-refinements, it yields the
theoretically predicated exponential rates of convergence. In this paper, we present
a novel approach to the solution of axisymmetric cavity problems which uses a
hierarchic H1 and H(curl) conforming finite element basis. A selection
of numerical examples are included and these demonstrate that the exponential rates
of convergence are achieved in practice.

Abstract:
The finite mass method is a purely Lagrangian scheme for the spatial discretisation
of the macroscopic phenomenological laws that govern the flow of compressible
fluids. In this article we investigate how to take into account long range
gravitational forces in the framework of the finite mass method. This is achieved
by incorporating an extra discrete potential energy of the gravitational field into
the Lagrangian that underlies the finite mass method. The discretisation of the
potential is done in an Eulerian fashion and employs an adaptive tensor product
mesh fixed in space, hence the name finite mass mesh method for
the new scheme. The transfer of information between the mass packets of the finite
mass method and the discrete potential equation relies on numerical quadrature, for
which different strategies will be proposed. The performance of the extended finite
mass method for the simulation of two-dimensional gas pillars under self-gravity
will be reported.

Abstract:
In this paper we examine the well-known magneto-quasistatic eddy current model for
the behaviour of low-frequency electromagnetic fields. We restrict ourselves to
formulations in the frequency domain and linear materials, but admit rather general
topological arrangements.
The generic eddy current model allows two dual formulations, which may be
dubbed E-based and H-based. We investigate so-called hybrid approaches that
combine both formulations by means of coupling conditions across the boundaries of
conducting regions. The resulting continuous and discrete variational formulations
will be discussed. Particular emphasis is laid on difficulties arising from the
topology of the conducting regions.

Abstract:
Many boundary integral equations for exterior Dirichlet- and Neumann boundary value
problems for the Helmholtz equation suffer from a notorious instability for wave
numbers related to interior resonances. The so-called combined field integral
equations are not affected. However, if the boundary $\Gamma$ is not smooth, the
traditional combined field integral equations for the exterior Dirichlet problem do
not give rise to an $\xLzwei{\Gamma}$-coercive variational formulation. This foils
attempts to establish asymptotic quasi-optimality of discrete solutions obtained
through conforming Galerkin boundary element schemes.
This article presents new combined field integral equations on two-dimensional closed
surfaces that possess coercivity in canonical trace spaces. The main idea is to use
suitable regularizing operators in the framework of both direct and indirect
methods. This permits us to apply the classical convergence theory of conforming
Galerkin methods.

Abstract:
Many boundary integral equations for exterior Dirichlet- and Neumann boundary value
problems for the Helmholtz equation suffer from a motorious instability for wave
numbers related to interior resonances. The so-called combined field integral
equations are not affected.
This article presents combined field integral equations on two-dimensional closed
surfaces that possess coercivity in canonical trace spaces. For the exterior
Dirichlet problem the main idea is to use suitable regularizing operators in the
framework of an indirect method. This permits us to apply the classical convergence
theory of conforming Galerkin methods.

Abstract:
We present a new variational direct boundary integral equation approach for solving
the scattering and transmission problem for dielectric objects partially coated with
a PEC layer. The main idea is to use to use the electromagnetic Calderon projector
along with transmission conditions for the electromagnetic fields. This leads to a
symmetric variational formulation which lends itself to Galerkin discretization by
means of divergence-conforming discrete surface currents. A wide array of numerical
experiments confirms the efficacy of the new method.

Abstract:
The mathematical foundation of a symmetric boundary element method for the
computation of eddy currents in a linear homogeneous conductor which is exposed to
an alternating magnetic field is presented. Starting from the $\mathbf{A}$-based
variational formulation of the eddy current equations and a related transmission
problem, the problem inside and outside the conductors is reformulated in terms of
integral equations on the boundary of the conductors. Surface currents occur as new
unknowns of this direct formulation. The integral equations can be coupled in a
symmetric fashion using the transmission conditions for the vector potential
$\mathbf{A}$ and the magnetic field $\mathbf{H}$. The resulting variational problem
is elliptic in suitable trace spaces.
A conforming Galerkin boundary element discretization is employed, which relies on
surface edge elements and provides quasi-optimal discrete approximations for the
tangential traces of $\mathbf{A}$ and $\mathbf{H}$. Surface stream functions
supplemented with co-homology vector fields ensure the vital zero divergence of the
discrete equivalent surface currents. Simple expressions allow the computation of
approximate total Ohmic losses and surface forces from the discrete boundary data.